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The connected door space is an enigmatic topological space in which every proper nonempty subset is either open or closed, but not both. This paper provides an elementary proof of the classification theorem of connected door spaces. More importantly, we show that connected door topologies can be viewed as solutions of the valuation $$f(A)+f(B)=f(A\cup B)+f(A\cap B)$$ f(A)+f(B)=f(A∪B)+f(A∩B) and the...
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